It has been proposed to use cellular materials for a wide range of structural, mechanical and thermal applications using both metal and non-metal materials. These cellular materials include spaces or voids embedded within a solid.
Cellular materials provide interesting combinations of physical and mechanical properties. One advantage provided by cellular materials involves light-weight construction. For example, cellular materials are capable of realizing a higher stiffness and a lower density for a given weight.
The relationship between the stiffness (S) of a flat panel, the Young's modulus (E) of the panel material (representing the ability of the panel material to resist elastic strain), and the thickness (h) of the panel can be represented by:S∝E×h3.  (1)Assuming that the panel maintains a constant footprint area and a constant weight, the relationship of the density (ρ) and the thickness (h) of the panel can be represented by:
                    h        ∝                              1            ρ                    .                                    (        2        )            Substituting Equation No. 2 for h in Equation No. 1 results in Equation No. 3.
                    S        ∝                              E                          ρ              3                                .                                    (        3        )            Cellular material that has imperfections such as the lack of uniformity and closure of the voids can be characterized by the following experimentally-found exponential relationship between the Young's modulus (E) and density (ρ).E∝ρ2.  (4)Substituting Equation No. 4 for E in Equation No. 3 results in Equation No. 5.
                    S        ∝                              1            ρ                    .                                    (        5        )            The following observations can be made for panels constructed of imperfect cellular material (i.e. material with nonuniform voids and/or lack of closure of voids). For a panel of a given footprint area and a given weight, constructing the panel to be twice as thick (2 h) (Equation No. 2) using imperfect cellular material half as dense (ρ/2) results in a panel that is twice as stiff (2S) (Equation No. 5). For a panel of a given footprint area, constructing the panel to be half as thick (h/2) using imperfect cellular material half as dense (ρ/2) results in a panel that maintains a given stiffness (S) and that is half the weight. Thus, imperfect cellular material provides certain benefits related to the relationship between density and stiffness as provided by Equation No. 5.
A perfect cellular material with uniform and closed voids is expected to provide a linear relationship between the Young's modulus (E) and the density (ρ), as represented by Equation No. 6.E∝ρ.  (6)Substituting Equation No. 6 for E in Equation No. 3 results in Equation No. 7.
                    S        ∝                              1                          ρ              2                                .                                    (        7        )            The relationship between the stiffness (S) and density (ρ) as expressed in Equation No. 7 is expected to be approached when the voids in the cellular material become more uniformly spaced and when a majority of the voids in the cellular material are closed voids.
The following observations can be made for panels constructed of material that approaches a perfect or ideal cellular material (material with uniform voids and closure of voids). For a panel of a given footprint area and a given weight, constructing the panel to be twice as thick (2 h) (Equation No. 2) using perfect cellular material half as dense (ρ/2) is expected to result in a panel that is four times as stiff (4S) (Equation No. 7). For a panel constructed of a given footprint area, constructing the panel to be one fourth as thick (h/4) using perfect cellular material half as dense (ρ/2) is expected to result in a panel that maintains a given stiffness (S) and that is ¼ of the weight. Thus, it is desirable to form cellular materials with uniform and closed voids.